On the book,categories for working mathematician, by S.Mac Lane, he gave a definition as follows: an additive category is an Ab-category which has a zero object 0 and a biproduct for each pair of its objects.However,the common definition is defined in the following way, A category C is called additive if satisfies the following conditions :
- Every Hom set is an abelian group;
- The composition of morphism satisfies two distribution laws;
- Exists zero object;
- The coproduct of finite objects exists.
By this definition, it's clear that additive category is defined on any abstract category, so why Mac Lane define it on an abelian groups category? Is it reasonable? There must be some reasons behind it . Just for simplicity or for some embedding theorem ? I am confused … Thanks in advance, any help will be greatly appreciated!